David M. McCandlish

Contingency and entrenchment in protein evolution under purifying selection

Significance How large a role does history play in evolution? Do later events depend critically on specific earlier events, or do all events occur more or less independently? If a change occurs early in evolution, does it become easier or harder to revert the change as time proceeds? Here, we explore these ideas in the context of protein evolution, by simulating sequence evolution under purifying selection and then systematically permuting the order of amino acid substitutions. Our results suggest that the amino acid substitutions that occur in evolution are typically contingent on the presence of prior substitutions, and that substitutions that occur early in evolution become entrenched and difficult to modify as subsequent substitutions accrue. The phenotypic effect of an allele at one genetic site may depend on alleles at other sites, a phenomenon known as epistasis. Epistasis can profoundly influence the process of evolution in populations and shape the patterns of protein divergence across species. Whereas epistasis between adaptive substitutions has been studied extensively, relatively little is known about epistasis under purifying selection. Here we use computational models of thermodynamic stability in a ligand-binding protein to explore the structure of epistasis in simulations of protein sequence evolution. Even though the predicted effects on stability of random mutations are almost completely additive, the mutations that fix under purifying selection are enriched for epistasis. In particular, the mutations that fix are contingent on previous substitutions: Although nearly neutral at their time of fixation, these mutations would be deleterious in the absence of preceding substitutions. Conversely, substitutions under purifying selection are subsequently entrenched by epistasis with later substitutions: They become increasingly deleterious to revert over time. Our results imply that, even under purifying selection, protein sequence evolution is often contingent on history and so it cannot be predicted by the phenotypic effects of mutations assayed in the ancestral background.

Modeling Evolution Using the Probability of Fixation: History and Implications

Many models of evolution calculate the rate of evolution by multiplying the rate at which new mutations originate within a population by a probability of fixation. Here we review the historical origins, contemporary applications, and evolutionary implications of these “origin-fixation” models, which are widely used in evolutionary genetics, molecular evolution, and phylogenetics. Origin-fixation models were first introduced in 1969, in association with an emerging view of “molecular” evolution. Early origin-fixation models were used to calculate an instantaneous rate of evolution across a large number of independently evolving loci; in the 1980s and 1990s, a second wave of origin-fixation models emerged to address a sequence of fixation events at a single locus. Although origin-fixation models have been applied to a broad array of problems in contemporary evolutionary research, their rise in popularity has not been accompanied by an increased appreciation of their restrictive assumptions or their distinctive implications. We argue that origin-fixation models constitute a coherent theory of mutation-limited evolution that contrasts sharply with theories of evolution that rely on the presence of standing genetic variation. A major unsolved question in evolutionary biology is the degree to which these models provide an accurate approximation of evolution in natural populations.

Visualizing fitness landscapes.

Fitness landscapes are a classical concept for thinking about the relationship between genotype and fitness. However, because the space of genotypes is typically high‐dimensional, the structure of fitness landscapes can be difficult to understand and the heuristic approach of thinking about fitness landscapes as low‐dimensional, continuous surfaces may be misleading. Here, I present a rigorous method for creating low‐dimensional representations of fitness landscapes. The basic idea is to plot the genotypes in a manner that reflects the ease or difficulty of evolving from one genotype to another. Such a layout can be constructed using the eigenvectors of the transition matrix describing the evolution of a population on the fitness landscape when mutation is weak. In addition, the eigendecomposition of this transition matrix provides a new, high‐level view of evolution on a fitness landscape. I demonstrate these techniques by visualizing the fitness landscape for selection for the amino acid serine and by visualizing a neutral network derived from the RNA secondary structure genotype–phenotype map.

The role of epistasis in protein evolution

Arising from M. S. Breen, C. Kemena, P. K. Vlasov, C. Notredame & F. A. Kondrashov 490, 535–538 (2012)10.1038/nature11510An important question in molecular evolution is whether an amino acid that occurs at a given site makes an independent contribution to fitness, or whether its contribution depends on the state of other sites in the organism’s genome, a phenomenon known as epistasis. Breen and colleagues recently argued that epistasis must be “pervasive throughout protein evolution” because the observed ratio between the per-site rates of non-synonymous and synonymous substitutions (dN/dS) is much lower than would be expected in the absence of epistasis. However, when calculating the expected dN/dS ratio in the absence of epistasis, Breen et al. assumed that all amino acids observed at a given position in a protein alignment have equal fitness. Here, we relax this unrealistic assumption and show that any dN/dS value can in principle be achieved at a site, without epistasis; furthermore, for all nuclear and chloroplast genes in the Breen et al. data set, we show that the observed dN/dS values and the observed patterns of amino-acid diversity at each site are jointly consistent with a non-epistatic model of protein evolution.

Epistasis and the Dynamics of Reversion in Molecular Evolution

Recent studies of protein evolution contend that the longer an amino acid substitution is present at a site, the less likely it is to revert to the amino acid previously occupying that site. Here we study this phenomenon of decreasing reversion rates rigorously and in a much more general context. We show that, under weak mutation and for arbitrary fitness landscapes, reversion rates decrease with time for any site that is involved in at least one epistatic interaction. Specifically, we prove that, at stationarity, the hazard function of the distribution of waiting times until reversion is strictly decreasing for any such site. Thus, in the presence of epistasis, the longer a particular character has been absent from a site, the less likely the site will revert to its prior state. We also explore several examples of this general result, which share a common pattern whereby the probability of having reverted increases rapidly at short times to some substantial value before becoming almost flat after a few substitutions at other sites. This pattern indicates a characteristic tendency for reversion to occur either almost immediately after the initial substitution or only after a very long time.

ON THE FINDABILITY OF GENOTYPES

Can we define a measure that describes how easy or difficult it is for a population to evolve to a specific genotype? For populations evolving under weak mutation on a time‐invariant fitness landscape, I argue that one appropriate measure is the expected waiting time, starting from equilibrium, for a population to become fixed for a given genotype. Under this definition for the “findability” of genotypes, I show that for any pair of genotypes (1) a population at equilibrium is always more likely to fix at the more findable before the less findable genotype and (2) the expected time to evolve from the more findable to the less findable genotype is always greater that the expected time to evolve in the opposite direction. Although increasing the fitness of a genotype always increases its findability, in general there is no simple relationship between the rank ordering of genotypes by fitness and the rank ordering of genotypes by findability. I also present a method for quantifying the relative contributions of mutation, selection, substitution rate, and probability of reversion to a genotypes findability.

Formal properties of the probability of fixation: identities, inequalities and approximations.

The formula for the probability of fixation of a new mutation is widely used in theoretical population genetics and molecular evolution. Here we derive a series of identities, inequalities and approximations for the exact probability of fixation of a new mutation under the Moran process (equivalent results hold for the approximate probability of fixation under the Wright-Fisher process, after an appropriate change of variables). We show that the logarithm of the fixation probability has particularly simple behavior when the selection coefficient is measured as a difference of Malthusian fitnesses, and we exploit this simplicity to derive inequalities and approximations. We also present a comprehensive comparison of both existing and new approximations for the fixation probability, highlighting those approximations that induce a reversible Markov chain when used to describe the dynamics of evolution under weak mutation. To demonstrate the power of these results, we consider the classical problem of determining the total substitution rate across an ensemble of biallelic loci and prove that, at equilibrium, a strict majority of substitutions are due to drift rather than selection.

Detecting epistasis from an ensemble of adapting populations

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite‐site fitness landscapes. First, we analyze the mean fitness trajectory—that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non‐epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics—the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions—can detect certain forms of epistasis in the underlying fitness landscape.

THE INEVITABILITY OF UNCONDITIONALLY DELETERIOUS SUBSTITUTIONS DURING ADAPTATION

Studies on the genetics of adaptation from new mutations typically neglect the possibility that a deleterious mutation might fix. Nonetheless, here we show that, in many regimes, the first mutation to fix is most often deleterious, even when fitness is expected to increase in the long term. In particular, we prove that this phenomenon occurs under weak mutation for any house‐of‐cards model with an equilibrium distribution. We find that the same qualitative results hold under Fishers geometric model. We also provide a simple intuition for the surprising prevalence of unconditionally deleterious substitutions during early adaptation. Importantly, the phenomenon we describe occurs on fitness landscapes without any local maxima and is therefore distinct from “valley crossing.” Our results imply that the common practice of ignoring deleterious substitutions leads to qualitatively incorrect predictions in many regimes. Our results also have implications for the substitution process at equilibrium and for the response to a sudden decrease in population size.

Mutation-biased adaptation in Andean house wrens

Genes, proteins, and genomes are pervasively shaped by biases in mutation that exert their influence by biasing rates of evolutionary change. Indeed, methods of phylogeny inference routinely assume that evolutionary rates will reflect transition–transversion bias and other common mutational biases. This influence is typically attributed to neutral evolution, presuming that mutation is a weak force easily overcome by selection, so that any noticeable effect of mutation must signal a lack of selection. However, in PNAS Galen et al. (1) propose a case of mutation-biased adaptation, in which the course of evolution reflects both a bias in mutation and a fitness benefit. How important are mutational biases in shaping adaptation? Are internal sources of orientation or direction in evolution—once dismissed as improbable (2)—more important than previously imagined?